3 3 votes Consider the matrix $A=\begin{bmatrix}4&1&0&0&1\\1&4&1&0&0\\0&1&4&1&0\\0&0&1&4&1\\1&0&0&1&4\end{bmatrix}$.The largest eigenvalue of $A$ is ______. Linear Algebra goclasses goclasses-cs-dpp goclasses-cs-dpp-day-244 linear-algebra goclasses-linear-algebra-practice-questions matrix eigen-value numerical-answers + – GO Classes 373 views answer comment Share Follow Print See 1 comment 1 1 comment reply akash_kumar 9 commented Apr 13 reply Follow flag similar type of pyq 0 0 replyShare Please log in or register to add a comment.
2 2 votes Row1 Sum = 4 + 1 + 0 + 0 + 1 = 6 Row2 sum = 1 + 4 + 1+ 0+ 0 = 6 Row3 sum = 0 + 1 + 4 + 1 + 0 = 6 Row4 sum = 0 + 0+ 1 + 4 + 1 = 6 Row5 sum = 1 + 0 + 0 + 1 + 4 = 6 when in matarix sum is constant then the largest eignen value = 6 akash_kumar 9 answered Apr 13 akash_kumar 9 comment Share Follow See 1 comment 1 1 comment reply GO Classes commented Apr 14 reply Follow flag Your idea is correct, but the conclusion needs one more step.Since every row sum is $6$, we get $A(1,1,1,1,1)^T = 6(1,1,1,1,1)^T$, so $6$ is an eigenvalue of $A$. However, this alone does not prove that $6$ is the largest eigenvalue. To prove that, we use the fact that for a nonnegative matrix, the largest eigenvalue is at most the maximum row sum. Here the maximum row sum is also $6$. Therefore no eigenvalue can be greater than $6$. Since $6$ is already an eigenvalue, it follows that the largest eigenvalue is $\boxed{6}$.Reference : https://web.pdx.edu/~caughman/Denny%20501%20final.pdf 4 4 replyShare Please log in or register to add a comment.
